1029 
7 M6 
py 1 



JNIVERSITY OF ILLINOIS BULLETIN 

Issued Weekly 
Vol. XX November 13, 1922 No. 11 

[Entered as second-class matter December 11, 1912, at the post office at Urbana, Illinois, under the 
Act of August 24, 1912. Accepted for mailing at the special rate of postage provided for in 
section 1103, Act of October 3, 1917, authorized July 31, 1918.] 



BULLETIN NO. 11 A ^ -r ^ 'lS 



i\^-^- 



BUREAU OF EDUCATIONAL RESEARCH 
COLLEGE OF EDUCATION 

RELATION OF SECTIONING A CLASS 

TO THE EFFECTIVENESS OF 

INSTRUCTION 

by 
Walter S. Monroe, Director 




PRICE IS CENTS 



PUBLISHED BY THE UNIVERSITY OF ILLINOIS, URBANA 
1922 



The Bureau of Educational Research was established by act 
of the Board of Trustees June 1, 1918. It is the purpose of the 
Bureau to conduct original investigations m the field of education, 
to summarize and bring to the attention of school people the results 
of research elsewhere, and to be of service to the schools of the 
state in other ways. 

The results of original investigations carried on by the Bureau 
of Educational Research are published in the form of bulletins. A 
complete list of these publications is given on the back cover of 
this bulletin. At the present time five or six original investigations 
are reported each year. The accounts of research conducted else- 
where and other communications to the school men of the state 
are published in the form of educational research circulars. From 
ten to fifteen of these are issued each year. 

The Bureau is a department of the College of Education. Its 
immediate direction is vested in a Director, who is also an instructor 
in the College of Education. Under his supervision research is 
carried on by other members of the Bureau staff and also by grad- 
uate students who are working on theses. From this point of view 
the Bureau of Educational Research is a research laboratory for the 
College of Education. 

Bureau of Educational Research 

College of Education 

University of Illinois, Urbana 



BULLETIN NO. 11 



BUREAU OF EDUCATIONAL RESEARCH 
COLLEGE OF EDUCATION 



RELATION OF SECTIONING A CLASS 

TO THE EFFECTIVENESS OF 

INSTRUCTION 



by 
Walter S^ Monroe. 



..•^■'^ 



PRICE IS CENTS 



PUBLISHED BY THE UNIVERSITY OF ILLINOIS, URBANA 
1922 



^^. 



LIBRARY OF CONGRESS 










PREFACE 

The educational experiment reported in this bul- 
letin was initiated by the former Director of the Bureau 
of Educational Research and the data collected under 
his supervision. The present Director of the Bureau 
is responsible for the tabulation of the data and for 
the preparation of this report. 

This investigation was made possible through the 
cooperation of Superintendent Peter A. Mortenson and 
of certain principals and teachers of the Chicago Public 
Schools. Not only did they cooperate in the collection 
of the data but they also made substantial contribu- 
tions to the project by supplying test materials. The 
writer is glad to acknowledge the indebtedness of the 
Bureau of Educational Research to all who contributed 
to this project. 

Walter S. Monroe, Director 
November 10, 1922 



Relation of Sectioning a Class to the 
Effectiveness of Instruction 

The problem. The purpose of this educational experiment was 
to determine the relative effect upon the achievements in certain 
school subjects of three plans of sectioning a class. A "class" is 
defined as the total number of children assigned to a teacher for in- 
struction even though they may be divided into two or more groups 
for instructional purposes. The three plans of sectioning a class 
considered in this investigation are: (1) teaching a class as a single 
unit; (2) dividing the class into two equal groups approximately 
equivalent with respect to general intelligence; (3) dividing the class 
into three equal groups approximately equivalent with respect to 
general intelligence. When a class is taught as one group, all of the 
pupils recite at the same time. Following the recitation there is a 
period for study. Thus under this plan the work of the teacher al- 
ternates between "hearing classes" and supervising the study of 
the pupils. When a class is taught as two sections, one group 
recites while the other group studies. In this case the teacher's 
time is almost wholly devoted to "hearing classes." Any supervi- 
sion of the study of the pupils is of necessity given incidentally and at 
irregular intervals when the teacher is fortunate enough to have a 
few minutes of leisure during a recitation period. When a class is 
divided into three sections, the conditions are much the same except 
that necessarily the length of the recitation periods is reduced. In 
general pupils of one section study during the recitation periods 
of the other two sections. 

The specific problem of this investigation was to determine 
the relative effect of these three plans of sectioning a class upon the 
direct results of instruction in certain school subjects. In other 
words this investigation sought to answer the question, "Which is 
the best plan of sectioning a class?" 

General plan of the experiment. If it were possible to secure 
three groups of classes so that all factors which affect the results of 
instruction were equivalent in the beginning of the experiment and 
. could be controlled throughout the experimental period, the simplest 
procedure would be to have one group of classes taught as a unit, 
another group taught in two sections and a third group in three sec- 



tions. However, it would be difficult, if not impossible, to secure 
exact equivalence of teaching ability and of pupil material. Our 
facilities for measuring the ability of teachers are extremely crude 
and at best it would be difficult to demonstrate that any differences 
found in the results of instruction were not produced largely by 
differences in teaching ability. It is true that we have a number of 
general intelligence tests which might be used to measure the quality 
of the pupil material. However, the limitations of these instruments 
are such that one would be unable to interpret small differences in 
the resulting achievements. 

In order to avoid these two difficulties this experiment was 
planned so that the same teacher should instruct a given class when 
organized according to two different plans of sectioning. This, 
necessarily, must be done during successive semesters. This proce- 
dure insured the constancy of the teacher, although not necessarily 
of teaching ability since the ability of a given teacher may vary 
from semester to semester with different types of class organization. 
In order that the pupil material might be the same for the two plans 
of class organization one hundred percent promotion was secured 
at the middle of the school year. Thus, a teacher who instructed 
a class as one section during the first semester of this experiment 
instructed the same pupils during the second semester but with the 
class divided into two or three sections. Other teachers taught 
classes organized according to other combinations of sectioning. 

This general plan of the experiment makes the semester a vari- 
able factor. It is possible that pupils may normally make greater 
progress during one semester than during the other. Furthermore, 
the gain of second trial scores over first trial scores is likely to be 
much greater than the gain of third trial scores over second trial 
scores simply because the pupils become acquainted with the testing 
procedure. In order to balance these two variable factors it was 
necessary to arrange experimental groups in pairs. Thus, corres- 
ponding to an experimental group of classes which was taught as a 
single section during the first semester and as three sections during 
the second semester, there was another group of classes taught as 
three sections during the first semester and as a single section during 
the second semester. In dividing a class into sections the scores 
yielded by the general intelligence tests were used to secure sec- 
tions of approximately equivalent pupil material. Six experimental 
groups of classes were organized as follows: 



Group I. Classes taught as a single section during the first 
semester and as three sections during the second semester. 

Group II. Classes taught as three sections during the first 
semester and as one section during the second semester. 

Group III. Classes taught "as one section during the first 
semester and as two sections during the second semester. 

Group IV, Classes taught as two sections during the first 
semester and as one section during the second semester. 

Group V. Classes taught as two sections during the first 
semester and as three sections during the second semester. 

Group VI. Classes taught as three sections during the first 
semester and as two sections during the second semester. 

So far as the writer knows, essentially the same methods of 
instruction and subject-matter were followed in all of these groups. 
The investigation was confined to Grades II, V, and VII in order to 
reduce the labor and expense. As these grades are fairly representa- 
tive of the three divisions of the elementary school, primary, inter- 
mediate and grammar, it is not likely that different results would be 
obtained in the other grades. The number of classes, the total en- 
rollment, and the number of complete records in each experimental 
group are given in Table I. 

TABLE I. NUMBER OF CLASSES, TOTAL ENROLLMENT, AND NUMBER 
OF COMPLETE RECORDS IN EACH OF THE EXPERIMENTAL GROUPS 



Grade 




I 


n 


Group 
III IV 


V 


VI 


Total 


II 


Number of classes'' 
Total enrollment 
Complete records 


7 
348 
240 


4 
201 
111 


3 
138 
103 


6 

288 
208 


7 
324 
224 


3 
162 
89 


30 
1461 

975 


V 


Number of classes 
Total enrollment 
Complete records 


2 
87 
70 


2 
92 

72 


8 
379 
326 


4 
192 
133 


4 
196 

157 


4 
181 
143 


24 
1127 
901 


VII 


Number of classes 
Total enrollijient 
Complete records 


3 
141 
119 


3 
140 
109 


5 
244 
186 


5 
214 
159 


2 
91 
86 




18 
830 
659 



The data collected. Through the cooperation of Superintendent 
Peter A. Mortenson of the Chicago Public Schools and of certain 
principals and teachers, the Bureau of Educational Research carried 
on this investigation during the school year of 1920-21. Experi- 



mental classes were organized in sixteen elementary schools.^ For 
measuring the general intelligence of the pupils the Pressey 
Primer Scale was used in the second grade, and the Illinois General 
Intelligence Scale in the other two grades. The achievements of the 
pupils in the second grade were measured by means of the Pressey 
Scale of Attainment No, 1. In the fifth and seventh grades achieve- 
ments were measured by Monroe's Standardized Silent Reading 
Tests, Revised, Monroe's General Survey Scale in Arithmetic, and 
Buckingham's Problem Scale in Arithmetic, Divisions 1 and 2. The 
general intelligence tests were given only at the beginning of the ex- 
periment, October 11, 1920. Form 1 of the achievement tests was 
given at this time. Form 2 of the achievement tests was adminis- 
tered at the close of the first semester, February 3, 1921. At the 
close of the experimental period, May 11, 1921, Form 1 was again 
given. 

The tests were administered by the teachers who also scored the 
test papers and entered the scores upon individual record cards. 
This, however, was done only after all of the teachers involved in the 
experiment had been called together for the purpose of acquainting 
them with the tests. In this explanation several tests were adminis- 
tered to the teachers in exactly the same way as they were to be ad- 
ministered to the pupils. In addition detailed instructions were 
supplied to the teachers for all steps of the work. Since no compari- 
sons were made between the scores yielded by tests administered by 
different teachers it is felt that this procedure in the administration 
of the tests does not seriously affect the results of the experiment. 

Limitations of the experiment to be kept in mind in interpreting 
the results. A number of conditions must be kept in mind in inter- 
preting the results. In the first place practically all of the teachers 
who cooperated in the investigation had been accustomed to teaching 
classes in two sections. A few, perhaps 1 in 20, had taught a class as 
a single section but, so far as the writer was informed, no teacher 
had had any experience in instructing a class in three sections. Thus, 
it is altogether likely that most of the teachers had acquired a techni- 
que of instruction which would prove more successful with a class 
divided into two sections than with a class divided into either one 
or three sections. Furthermore, there appears to be a prejudice 



^These sixteen schools were the following: Brown, Dante, Douglas, Fiske, Jenner, 
Julia Ward Howe, Morse, Otis, Pullman, Scanlan, Shields, Spry, Van Vlissingen, Ward, 
Wentworth, and West Pullman. 



against the division of a class into three sections. Thus, there is 
introduced a factor which may be expected to produce greater achieve- 
ments in classes taught as two sections than in classes taught as 
either one or three sections. The effect of this factor is, however, 
unknown but it should by all means be recognized in interpreting 
the results. 

The instruments used for measuring the achievements of the 
pupils do not measure all achievements resulting from instruction. 
They can be considered to do no more than measure representative 
samples of the achievements within their respective fields. Outside 
of silent reading and arithmetic, in which tests were given, there are 
many important achievements of which no attempt was made to 
secure direct measurements. It is, of course, possible that the 
measures of achievements secured correlate closely enough with all 
other achievements resulting from instruction, that a sufficiently 
accurate index of all achievements is furnished for judging the re- 
lative effectiveness of the instruction in the different experimental 
groups. However, convincing experimental evidence on the point 
is wanting and, for this reason, due caution must be exercised in 
extending the conclusions of this experiment to school subjects other 
than silent reading and arithmetic, as well as to the more subtle 
outcomes engendered by the social contacts of the school room. 

Finally, it must be remembered that this investigation was 
carried on in classes enrolling approximately 45 pupils. Hence 
it does not necessarily follow that the conclusions would apply to 
classes enrolling 20 to 30 pupils. It is possible that this change in 
the size of class might produce a complete reversal in the conclusions. 

Method of summarizing data. After rejecting records which 
were incomplete and obviously inaccurate, the scores yielded by an 
application of a test were combined in a total distribution for each 
experimental group. Thus, a distribution was formed of the first 
trial scores made on Monroe's Standardized Silent Reading Tests, 
Revised, by the group of fifth grade pupils enrolled in "classes taught 
as a single section during the first semester and as three sections dur- 
ing the second semester." In the same way distributions of scores 
were formed for each of the experimental groups and for each appli- 
cation of the test. The gain in achievement during the first semester 
was found by subtracting the average score for the first trial of a 
test from the average score of the second trial. The gain for the 
second semester was found by subtracting the average score of the 



second trial from that of the third trial. A second measure of gain 
was secured by following a similar procedure with the median scores 
but these gains are not given in this report as they were, in general, 
in agreement with those calculated from the average scores. 

In calculating these gains no account was taken of the possible 
non-equivalence of the different forms of the tests used. In fact no 
accurate information concerning the equivalence of duplicate forms 
is available except for Monroe's Standardized Silent Reading Tests, 
Revised, and for Monroe's General Survey Scale in Arithmetic. 
The duplicate forms of these two tests have been shown to be approx- 
imately equivalent,^ However, since Form 1 of each test was used 
twice and the average scores calculated frt)m it were used both as 
subtrahends and minuends, and since the gain for any plan of section- 
ing is computed from both semesters the non-equivalence of Forms 1 
and 2 of the tests used will not affect the comparisons of gains made 
in the following tables. 

The point scores yielded by the different tests are expressed in 
terms of different units and from different zero points. Thus before 
any combination from the results of the different tests can be made 
it is necessary to express the gains in terms of a common unit. The 
usual assumption in such cases is that the standard deviation of the 
distribution of scores represents the same increment of ability for 
one test as for another. On the basis of this assumption a total dis- 
tribution for each test was secured by adding the distributions of 
the six experimental groups within a grade. This was done for the 
scores secured at each period of testing. The average of the three 
standard deviations was assumed to represent the same increment of 
ability for each test and was used as a divisor for reducing the gains 
to the basis of a common unit. ' For example, during the first semester 
the fifth grade pupils in Group I classes made a gain in arithmetic of 
23.82 points. During the second semester they made a gain of 21.5 
points. The average standard deviation of the arithmetic scores 
in the fifth grade is 19.65. Using this as a divisor we secure as 
quotients 1.21 and 1.09. In this manner the entries in Tables II, 
III and IV were obtained. The two quotients whose calculation was 
explained are given in Table III. 

Tables II, III and IV are similar in structure and are to be read 
in the same way. The gains for the different experimental groups 

^Monroe, W. S. Illinois Examination, University of Illinois Bulletin Vol. 19, No. 
9, Bureau of Educational Research Bulletin No. 6, Urbana: University of Illinois, 
1921. 70 p. 

10 



S o 






< 


Q 




s 


i 


o 


s"^-. 


g ^_ <M 00 


"^ CO 

g ^ -, 0C3 


■ c 
"5 


30^.8 


3:22 s 


CO -^ <N 




Q 


2 


CO 


s 


.S 

o 


— < 00 u-i 

3 f". ^. =^ 


r- ^ ■* 


3^5^ 


'5 

o 


r- 00 00 

;::;- CO "^ o\ 


c=~^". s^ 


f:j- vi CO ON 


CO 


Q 


^ 


;* 


8 

r 


c 
O 


r- r- <N 

?o ^ -: r^ 


^ VO 
^ Lo <N 00 


t^ VO (S 
g<N CO 00 


c 

O 


St^S. 


s°^.^. 


g^ ^. °° 


1 


Q 


u-i 


^ 

r 


CO 
CO 




s-.^^ 


, s"-2^ 


^ vo <N OS 

3^ t ^. 


c 


3^2." 


32:^, s 


CO '-^ CN 


H 


Q 


S 





CO 

r 


c 
O 


>ri .^ CO 

5 "^ ^. °°. 


01 LO -* 

g"^ CO a^ 


s":-s 


c 
O 


£^. ^.^ 


^^ r^ On 

;2r —; as OS 


S ^— ^ 


"c 
^ 


1 




§i 






a, 

5 


< 


<! 


a, 

< 



^ o 



< 


^^ 






'Z, 


w 


u 


k; 


'p 


o 






u. 


C/3 


0. 


u 


x 


o 


u 




H 


U 


^ 


> 


u 


<J 


p; 




u 


>! 


1*, 


n 


u< 


(v; 


Q 


(^ 


rii 


Q 


S 


^ 


y 


a, 




^ 


w 

Q 


O 


< 


C/5 


>■ 


:z: 






H 


<; 


a: 


o 


w 


^' — ' 


^ 




w 




> 




w 








K 




o 




<; 




Iz; 





< 


Q 


-^ 2 iS 


O 


P^ovon ^u^cs-* j:^>ric-5-^ 




^2^^ ^^^^^ ^^^,^, 


i § 

s •> 

.- c 

< 


Q 


^ ^ 8 


c 
O 




c 


;l^vo^(N Qoogjun ^r-o-i; 




Q 


CO r- ^ 

o -^o -« 

1 


c 
O 


coot^ oo^^f^ OOW-, 

^^^c4 ^^oo j:^oir-^ 


O 


3^^^ 3^^^ 3^^^ 


s 

< 


Q 


2 S ^ 

r i' 


.S 
'5 
O 


fo 'R '^. '^. 3 "^ ~ "^ g <^. '^, <^ 




^CNCMt-- ^VO^'O ^U^CN^ 


c 
.2 

6 


Q 


(~) >-n ^j^ 
S ^ <N 

r 


c 


^r^-** O'^-'f vo<r5>^ 


c 


-0 00-* OsOO w-icoON 

;:i;~co'^'^ ^(Ncoco jq-Tt<r^Lo 


C 

1 


Q 


f^ ^ t 

r 


c 
O 


OnOs-J^ n^O-"*" CTsrocO 
^oo-- ^oo^o ^sor^r^ 

■^ r ■ i' "-" 1 ^ 1 


c 

"a 


oow^cs CNt^cO COCO<0 

Q\o'j-iM3 ^^r—'^^f^ j::j-(NO'o 


Z 0^ 


O (N vo CO r- CO 

r-- r- CN CO W-, rt< 

CO ^ ^ ■^ 


c 


J 
) 


iJ 11 <u 

em M tifl 

^-^S s>2 -^rS 

■^■-hS n^4J ?^>-5 

< < < 



* 




w 




o 




< 




ai 




O 




X 




H 




^ 




(d 




> 




W 




Cfl 




U 




ac 




H 




!?; 








c/1 




a. 




!3 




O 




oi 




O 






w 

8 

C/3 






c^ 


w 


w 


CJ 




< 


X 


c^ 


m 


w 


H 
^ 


> 


u 


■? 


c< 


n 




oi 




(^ 



5 8 

s a 
w 



< 


(5 


■<t< 10 


c 


PJ^ponco ^io»/^«7l 






_1> o 

2 ■> 
-S "° 

< 


Q 


r 




r^ 00 00 c-5 00 vo 
g -: t^. ^ g 


c 
O 


^^ r^ r^ (N CO c^ 00 
■^ n <^ oi "^ rn c^ c~\ 


1 § 
< 


Q 


r r 


.S 

o 


00 00 TfH CO 0^ 
g ^ (N g '^. <N 






B 

< 


Q 


tS 00 


c 



Q^^^ gsj^s^. 


.S 

a 


Tji rt CO ^ ^ rt 




(5 


1 1 





j::^- 0^ CN >^ ^ ■* ON \o 


c 






.s 

1 


Q 


10 

8 , - ?) 
1 1 


-S 





.s 

to 




.-H U^ Tt< «0 VO 




0\ a\ vo On 


c 


I 

5 


I 
II 

Average 

III 
IV 

Average 



are arranged in pairs. In Table II, the gain for Group I on Test 1 
when taught in classes of one section is 1.42. When taught in three 
sections the gain is .55. The gain for Group II classes when taught 
in one section is .90 and when taught in three sections it is 1.11. 
The Group I classes were taught in one section during the first 
semester but the Group II classes were taught in one section during 
the second semester. This difference in time is largely responsible 
for the differences in the size of the gains. 

Interpretation of results. In interpreting the gains in Tables 

II, III and IV it is necessary to keep in mind both the constant and 
variable errors of measurement which are involved in the original 
data as well as the chance variations in tlte gains due to sampling. 
The variable errors of measurement in the original data depend upon 
the reliability of the tests used. If we assume a coefficient of re- 
liability^ of .84 for Test 1, it can be shown that the probable variable 
error of measurement is approximately .25 when expressed in terms 
of sigma which is the unit used in expressing the gains in Tables II, 

III, and IV."^ A probable error of measurement of .25 means that 
the scores for 50 percent of the pupils involve variable errors which 
are less than .25. For the other 50 percent the variable errors will 
be greater than ,25. The presence of variable errors of measurement 
affects the average of the scores as shown by the following formula 
in which N is the number of scores upon which the average is based. 

P.E.M 



P.E. 



M average — 



Substituting in this formula for Group I, we find the probable error 
of measurement of the average (P. E. m average) is .017; for Group II 
it is .024. The gain 1.42 is the difference between the two averages. 



^The coefficient of reliability assumed here is probably higher than would be found 
for this test. When based upon the scores of a single grade, the coefficient of re- 
liability for Monroe's General Survey Scale in Arithmetic is approximately .85. For 
Monroe's Standardized Silent Reading Test 1, Revised, the coefficients of reliability 
are approximately .75 for rate and .65 for comprehension. For Test II they are 
about .08 higher. The reliability of the other tests is not known. 

*The formula for the probable variable error of measurement is 
P.E.M = .6745 ai/iH^^ 
In this case a = l. 



14 



The probable error of the difference of the two averages is given by 
the following formula 



^V 



-2 2 



P.E.Dif.=\p.E.i + P.E.2 

In this formula P. E.i and P. E.2 stand for the probable errors of 
measurement of the two averages whose difference is taken. In this 
case P. E.i is equal to P. E.2 since we have used the average of the 
standard deviations of the several distributions in reducing the gains 
to a comparable basis. Applying the above formula, we find that 
the probable variable error of measurement to be associated with 1.42 
is .024 and with .90 is .034. The formula for the probable error of 
the sum of the two averages is the same as that for their difference. 
Hence we may calculate the probable error of measurement to be 
associated with the average gain 1.16 by taking one half of the 
probable error of measurement of the sum of the two averages. The 
P. E.M of the average gain 1.16 is .020. 

Since the probable variable error of measurement depends only 
upon the magnitude of the standard deviation of the scores and the 
number of scores, we will obtain the same result for the gains of these 
two groups when taught in classes of three sections. The probable 
variable error of measurement of the difference (.33) may be calcu- 
lated by the formula given above. It is .028, 

This probable variable error of measurement is relatively small 
in comparison with the gain .33, and in general when an average or 
difference is three or four times its probable error it can be considered 
significant. Hence, if we had to consider only the variable errors of 
measurement we would be justified in asserting that this difference 
was significant and could not be due to the presence of these errors 
in our original data. However, it should be remembered that we 
have been liberal in the estimate of the coefficient of reliability. 
It is likely that the true value of the probable error is much 
larger. 

Since all gains are expressed in terms of a common unit the prob- 
able variable^errors of measurement found for the entries under Test 1 
will apply also to Tests 2, 3, and 4 provided we assume the same co- 
efficient of reliability for these tests. The probable variable error of 
measurement of the average is affected by the number of cases from 
which the average is computed. Hence for the gains made by other 
groups it will be slightly greater, since the number of scores is smaller 



for those groups. In Table HI the number of scores in Groups III 
and IV is slightly larger. Hence a smaller probable variable error 
of measurement will be found, but for all of the other groups it will 
be larger than the one which we have considered in detail. In several 
cases the difference in gains is so small that when compared with the 
probable variable error of measurement it cannot be considered as 
significant. 

In addition to the variable errors of measurement, it is necessary 
to consider the chance variations in the gains due to sampling even 
when the sample has been chosen without bias. The probable error 
of an average due to sampling is given by the following formula 

Cdist. 

P. E.s = .6745-^^ 

Since sigma (a) has been used as a unit in terms of which the gains 
are expressed, Cdist. equals 1 for our calculations.^ In the case 
of Group I, P. E.s=.044. The gain 1.42 is the difference between 
two averages and hence it would be necessary to apply the formula 
for the probable error of the difference of the two averages. This 
being done we find that the P. E.s to be applied to the gain 
(1.42) is .062. In case of Group II, P. E.s=.064 and for the differ- 
ence between the two averages it is .090. For the average 1.16, 
P. E.s = .055. For the difference .33, P. E.s = .078. 

When we consider the probable error due to sampling (.078) 
in addition to the probable variable error of measurement (.028) the 
difference (.33) would probably be significant and indicate a slight 
superiority in achievement as measured by Test 1 for the pupils 
taught in classes of one section, provided no other errors could be 
considered to affect this difference. It is, however, necessary to 
consider the constant errors of measurement. Their exact magni- 
tude can not be known but their presence is evident. For example, 
in Table II the gains on Test 1 for Groups I and II when taught as 
one section are 1.42 and .90 respectively. The gain of 1.42 was made 
during the first semester and is the difference between the first and 
second trial scores. The gain of .90 was made during the second 
semester and is the difference between the second and third trial 
scores. Due to the pupils becoming acquainted with the tests and 

^This is not the true value of a. The variable errors of measurement tend to in- 
crease the value of the obtained sigma. The relation is given by the formula 

0'true = 0' obtained K rjj 

16 



the testing procedure, both of these gains involve a constant error. 
This tends to make the obtained gain larger than the true gain, but 
as the practice effect of the second trial scores over the first trial 
scores is larger than that of the third trial over the second trial scores, 
it is reasonably certain that the gain for Group I (1.42) contains the 
larger constant error. The gains made by these two groups when 
taught in classes of three sections are .55 and 1.11. Both of these 
gains involve a constant error but in this case the larger constant 
error is found in the gain for Group II. Each of the average gains 
for these two groups (1.16 and .83) includes a relatively large constant 
error but the two errors are much more nearly equal than those in- 
cluded in the gains for each group separately. Hence, we are probably 
justified in considering their difference (.33) to be relatively un- 
affected by the presence of constant errors in any of our original data. 

However, the neutralization of the constant errors which seems 
plausible, if not probable, in the case we have just considered does 
not appear to have taken place in a number of the other differences 
in this group of tables. With the exception of Groups I and II in 
Table II some of the differences are positive but others are negative 
for each pair of groups, although it is not impossible that a given plan 
of sectioning a class might be more effective in one subject than in 
another. The variations in the signs of the differences do not appear 
to occur in such a way as to justify this explanation of the negative 
gains. It is likely that a constant error was introduced in certain 
groups of scores which was not neutralized in the difference. For 
example. Group VI is shown by Test 2 to have made a larger gain 
during the second semester when taught in two sections. Each of 
the other tests shows a smaller gain for this semester and this we 
should expect as the gain is the difference between the second and 
third trial scores. The probable explanation of this condition is that 
in some way a constant error was introduced in one set of scores yield- 
ed by Test 2 for Group VI. An examination of Tables III and IV 
reveals several similar instances. Hence, we are forced to the con- 
clusion that at least certain sets of scores involve an unknown con- 
stant error. The fact that this happened in certain cases tends 
to make one suspicious of the presence of an unknown constant error 
in other sets of scores even though evidence of its presence is lacking. 

It is perhaps significant that in the case of the differences in 
gains between classes taught as one section and classes taught in 
three sections, eight gains are positive while six are negative. The 

17 



same situation prevails with respect to the gains made by classes 
taught in one section when compared with the gains made by classes 
taught in two sections. For classes taught in two sections compared 
with classes taught in three sections, we have records only in the 
second and fifth grades. Four of the differences are positive while 
five are negative. 

Conclusion. The facts presented in Tables II, III, and IV and 
the errors they include appear to justify the conclusion that there is 
no evidence of greater achievements being made by pupils when 
taught in classes organized on the basis of one plan of sectioning than 
in classes organized on a different plan of sectioning. Since the teach- 
ers were more experienced in teaching classes in two sections and 
probably preferred this plan of organization this condition might 
appear to mean that the division of classes into two sections was the 
least efficient of the three plans. However, in the writer's judgment 
this conclusion is not justified. The most obvious inference, in his 
opinion, to be drawn from the data of this experiment is that the 
educational tests used do not yield sufficiently accurate and precise 
measures of achievement to make possible the determination, under 
the conditions of this experiment, of the best method of sectioning 
a class. It is likely that the differences in the gains made during a 
period of less than a semester are not large. This being the case it is 
necessary either to extend the experimental period or to secure more 
precise measures of achievement. The magnitude of the probable 
variable error of measurement of the difference and also of the prob- 
able error due to sampling can be decreased by increasing the number 
of pupils in the experimental groups, but the constant errors are not 
affected by any increase in the number of cases. Certain constant 
errors are neutralized in the differences but, as we have shown, other 
constant errors which occur in only certain sets of scores were not 
eliminated. The presence of these constant errors is due to imper- 
fections in the educational tests used. Therefore, it appears that 
until our instruments for measuring achievements of school children 
are materially improved we cannot expect such educational ex- 
periments as the one described in this report to lead to reliable 
conclusions. 



THE UNIVERSITY OF ILLINOIS 

THE STATE UNIVERSITY 

URBANA 

DAVID KINLEY, Ph.D., LL.D., President 



The University Includes the Following Departments 

The Graduate School 

The College of Liberal Arts and Sciences (Ancient and Modern Languages 
and Literatures; History, Economics, Political Science, Sociology, Philosophy, 
Psychology, Education; Mathematics; Astronomy; Geology; Physics; Chemistry; 
Botany, Zoology, Entomology; Physiology, Art and Design) 

The College of Commerce and Business Administration (General Business, 
Banking, Insurance, Accountancy, Railway Administration, Foreign Commerce; 
Courses for Commercial Teachers and Commercial and Civic Secretaries) 

The College of Engineering (Architecture; Architectural, Ceramic, Civil, Elec- 
trical, Mechanical, Mining, Municipal and Sanitary, Railway Engineering, and 
General Engineering Physics) 

The College of Agriculture (Agronomy; Animal Husbandry; Dairy Husbandry; 
Horticulture and Landscape Gardening; Agricultural Extension; Teachers' 
Course; Home Economics) 

The College of Law (Three-year and four-year curriculums based on two years 
and one year of college work respectively) 

The College of Education 

The Curriculum in Joxhinalism 

The Curriculums in Chemistry and Chemical Engineering 

The School of Railway Engineering and Administration 

The School of Music (four-year curriculum) 

The Library School (two-year curriculum for college graduates) 

The College of Medicine (in Chicago) 

The College of Dentistry (in Chicago) 

The School of Pharmacy (in Chicago; Ph.G. and Ph.C. curriculums) 

The Summer Session (eight weeks) 

Experiment Stations and Scientific Bureaus: U. S. Agricultural Experiment 
Station; Engineering and Experiment Station; State Laboratory of Natural 
History; State Entomologist's Office; Biological Experiment Station on Ilbnois 
River; State Water Survey; State Geological Survey; U. S. Bureau of Mmes 
Experiment Station. 

The library collections contain May 1, 1922, 523,230 volumes and 120,151 pam- 
phlets. For catalogs and information address 

THE REGISTRAR 

Urbana, Illinois 



JLLETINS OF THE BUREAU OF EDUCATIONAL RE- 
SEARCH, COLLEGE OF EDUCATION, UNIVERSITY 
OF ILLINOIS, URBANA, ILLINOIS 

Price 

No. 1. Buckingham, B, R. Bureau of Educational Research, 

Announcement, 1918-19 IS 

No. 2. First Annual Report **. 25 

No. 3. Bamesberger, Velda C. Standard Requirements for 

Memorizing Literary Material 50 

No. 4. Holley, Charles E. Mental Tests for School Use. 

(Out of print) .50 

No. 5. Monroe, Walter S. Report of Division of Educational 

Tests for 1919-20 25 

No. 6. Monroe, Walter S. The Illinois Examination 50 

No. 7. Monroe, Walter S. Types of Learning Required of 
Pupils in the Seventh and Eighth Grades and in 
the High School 15 

No. 8. Monroe, Walter S. A Critical Study of Certain Silent 

Reading Tests 50 

No. 9. Monroe, Walter S. Written Examinations and Their 

Improvement. (In preparation) 50 

No. 10. Bureau of Educational Research. Relation of Size 

of Class to School Efficiency 50 

No. 11. Monroe Walter S. Relation of Sectioning a Class to 

the Effectiveness of Instruction .15 



